The Computational Beauty of Nature — Spring 2020

Problem Set 1: Complex Numbers and Fractals

Due by class time Thursday, February 13

Reading and Video

• Watch the documentary Fractals: the Colors of Infinity (54 minutes), narrated by the science-fiction writer Arthur C. Clarke. This was released in 1994, so the video and audio quality unfortunately is rather low, but the film nevertheless gives an excellent overview of the Mandelbrot set and fractal geometry.

• For next week, read the following Chapters of The Computational Beauty of Nature by Gary William Flake:

  • Chapter 5 (Self-Similarity and Fractal Geometry, pages 61-76)
  • Chapter 6 (L-Systems and Fractal Growth, pages 77-92)

Written Exercises

For these exercises, you should write up your solutions either by hand or using a word-processing program such as Word. Handwritten work is acceptable as long as you write neatly and legibly, but you'll lose points if I have to struggle to read your answers. For your numerical computations, you may use a calculator or this website for help with your complex number arithmetic (squaring, adding, etc.), but your written answers must clearly show each of the steps involved in your computations.

1. Calculate the magnitude of the complex number -6+8i. Show clearly how you computed your answer.

2. Is the number -2 a member of the Mandelbrot set? Demonstrate clearly why or why not.

3. Is the number 0.5 a member of the Mandelbrot set? Demonstrate clearly why or why not.

4. Is the number i a member of the Mandelbrot set? Demonstrate clearly why or why not.

5. Is the number 0.3+i a member of the Mandelbrot set? Demonstrate clearly why or why not.

6. Use Newton's method to calculate the square root of 4, starting from an initial guess of x = 1. Show the equation you are iterating, and the intermediate values of x at each step of the process. Then repeat the calculation starting from an initial guess of x = -1.5, showing your work. Do you get the same answer as before?

7. Use Newton's method to calculate the square root of i, starting from an initial guess of x = 1. Show the equation you are iterating, and the intermediate values of x at each step of the process. Then repeat the calculation starting from an initial guess of x = 1+i, showing your work. Do you get the same answer as before?

8. The Julia set for c = 0 is shown below (the black region). It is a perfect circle. What is the radius of this circle? Give a clear explanation of why the Julia set is a circle, and why it has that particular radius.